Suppose that $V$ is a $K-$vector space. For each positive integer $k$ the set $G_k(V )$ := {$l ⊂ V$| $l$ is a $k$-dimensional linear subspace} is called the Grassmann manifold of $k$-planes in $V$ . Assume that $V=K^n$ and define an atlas on $G_k(V)$ as follows. Let $e_1,...,e_n$ be the standard basis for $K^n$. Each partition {1,2,...,n} = I ∪ J, I = {$i_1 < ··· < i_k$}, J = {$j_1 < ··· < j_{n−k}$} of the first $n$ natural numbers determines a direct sum decomposition $K^n =V =V_I ⊕V_J$ via the formulas $V_I= Ke_{i_1}+ ··· + Ke_{i_k}$ and $V_J = Ke_{j_1}+ ··· + Ke_{j_{n−k}}$. Let $U_I$ denote the set of $l ∈ G_k (V )$ which are transverse to $V_J$, i.e. such that $l ∩ V_J$ = {$0$}. The elements of $U_I$ are precisely those k-planes of form $l=graph(A)$ where $\color{blue} {A:V_I →V_J}$ is a linear map. Define $φ_I :U_i →K^{k×(n−k)}$ by the formula $φ_I(l)=(a_{rs})$, $Ae_{i_r}= \sum_{s=1}^{n-k} a_{rs}e_{j_s}.$
My question is that what exactly the linear map $A$ is, since one can define many linear maps between two linear spaces. Thanks in advance
The point is that given any linear map $A$, we have a corresponding specific element of $U_I$, and vice versa. So every linear map gives you a different element, and which linear map it can be is not restricted.